I'm stuck on a question which asks me to find the period of oscillation of a cylinder of radius r rolling inside a tube of internal radius R.

The part which is throwing me is the part of the question that asks to find a relation between ω (the angular velocity of the cylinder) and θ (the angle the centre of mass of the cylinder makes with the vertical).

The answer is supposedly: ω = θ-dot * (R-r)/r

the problem I have is no matter how I think about it, I can't get that answer. When I think about the cylinder rolling (it doesnt slip) along some distance of the inside of the tube, it goes through some angle Ø of rotation. The distance it has travelled I think is rØ, which should equal the distance along the inside of the tube: Rθ. (this is taking ω = Ø-dot)